4. Solve the following boundary value and initial value problem for wave equation Utt-c²Uzz = 0,0 0 U(0, t) = 0, U(T, t) = 0 U(x,0)=bsin x + 2b sin(2x), Ut(x, 0) = sin x
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- Solve the inhomogeneous wave equation on the real lineUtt − c2Uxx = sin x, x ∈ RU(x, 0) = 0, Ut(x, 0) = 0.Explain what theory you are using and show your full computations.Consider the wave equation 0 0, with u(0,1) = 1(xt)= 0, u(x,0) = sin x and =0 at t=0. Then u isShow that the function Z = sin(wct)sin(wx) satisfies the wave equation
- Let f(x,t)=cos(12x+4t). Find the value of K so that f satisfies the wave equation ∂^2f/∂x2=K∂^2f/∂t^2for wave equation, seperation of vairables u(x,t)=X=(x)T(t)= Use variables separation method to solve the wave equation uxxutt. This function is defined on spatial domain 0 0. Subject to boundary conditions: ux(0, t) = u,(a, t) = 0 and initial conditions: u(x, 0) = 0 and u₁(x,0) = f(x)
- 2. The position vector of a particle is given by r(t)= (2 cos t sin t)i +(cos^2 t - sin^2 t)j + (3t)k If the particle begins its motion at t = 0 and ends at t = pi, find the difference between the length of the path traveled and the distance between start position and end positionPlease short steps and final answer. Solve the wave equation using Fourier Transform: o'u ou - c0 0 %3D -12x utx, 0) = H(x)e %3D u,(x, 0) = 0. Oa. u (x, t) = Real sin wt (12+iw) %3D 2 я Ob. 1 iwx u (x,t) Real cos wt dw (12+iw) Oc. iwx u (x,1) Real sın wt %3D w (12+iw) Od. 1 u (x,1) = Real cos wt w (12+iw) Oe correct ancueSolve the following wave equation using finite difference method. 4fxx = ftt - Given: f(0, t) = 0 and f(1, t) = 0 f(x, 0) = ft(x, 0) = 0 sin(x) + sin(2x) (Ref: Hyperbolic Equation)
- Find the solution to the wave equation on half-line: Utt=C²Uzr x > 0, t > 0, u(0, t) = 0, t> 0, u(x,0)=1/x, u₁(x,0) = e, x > 0.Which of the following is most suitable solution for wave equation = c2? %3D at2 ax2 .(a) y = (AeP* + Be-P*)(Ce pt + De ept) (b) y = (Acos px + Bsin px)(Ccos cpt + Dsin cpt) (c) y = (Ax + B)(Ct + D) (d) y = (Aepx + Be-px)(Cep*t + De-ep*r) O a O b O c O d3. Solve the radial wave in R3 with initral data gler) =4-r², Yer)=0 equation = Au U, =